Beta-expansion and continued fraction expansion over formal Laurent series

Let x ∈ I be an irrational element and n 1, where I is the unit disc in the field of formal Laurent series F((X−1)), we denote by kn(x) the number of exact partial quotients in continued fraction expansion of x, given by the first n digits in the β-expansion of x, both expansions are based on F((X−1)). We obtain that lim inf n→+∞ kn(x) n = degβ 2Q∗(x) , lim sup n→+∞ kn(x) n = degβ 2Q∗(x) , where Q∗(x),Q∗(x) are the upper and lower constants of x, respectively. Also, a central limit theorem and an iterated logarithm law for {kn(x)}n 1 are established. © 2007 Elsevier Inc. All rights reserved.